Category Archives: 我與A.Maths爭鬥三十年

1.1.6 What is Mathematics?

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As long as you can realize the difference between analytic statements and synthetic statements, you can know what pure mathematics is.

Pure Mathematics is a system of useful analytic statements.[10]

Pure Mathematics is a system of useful tautologies, whether obvious or unobvious.

In Physics, every generation of physicists have to update the previous theories. For example, Einstein’s theory of gravity has updated Newton’s, explaining what Newton’s gravitation could not explain. But for Pure Maths, although every generation of mathematicians also create new mathematics, the new theories do not and cannot contradict the old ones. For example, “1+1=2” is always true, even in thousands of years later.[11]

1.1.7 Why maths is always true but physics is not?[12]

Pure Maths is a system of analytics statements. Analytic statements say nothing about the world. When you say nothing, you cannot be wrong.

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[10] In Philosophy of Mathematics, this is called the Formalist’s theory of Mathematics. There is a bug in the formalist’s system. It is about the status of the axiom of infinity. For reference, see Bertrand Russell’s Introduction to Mathematical Philosophy.

[11] Mathematics is eternal, as it is timeless, or outside time.

[12] Mr. Lee

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2008.05.20 Tuesday \copyright CHK^2

1.1.5 Contrast

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Table 1.1: Contrasting

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Analytic

Synthetic

Logic
Pure Mathematics

Applied Mathematics
Physics

Relations of ideas

Matter of fact[6]

Deduction

Induction

Say nothing

Say something[7]

Always correct

Maybe wrong

Theory

Experiment

Software

Hardware

Computer Science

Computer Engineering

Mathematical Geometry

Physical Geometry

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[6] David Hume
[7] about the physical world

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2008.05.17 Saturday \copyright CHK^2

1.1.2 Logic and Pure Mathematics

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Pure Mathematics is a system of (nontrivial) tautologies. Roughly speaking, a tautology[2] is an analytic statement.

For example, consider this mathematics statement:

2 + 2 = 4

You do not have to do any kind of real world experiments in order to verify the statement. As long as you know the meanings of the symbols “2”, “+”, “=”, and “4”, you know that the statement is correct, and always. Of course, it says nothing about the physical world.

In pure mathematics, since you cannot and do not have to say anything about the real physical world, you can do[3] anything you like. Just like what you do when designing the rules of chess.[4] You can do anythings as long as they are

consistent and

interesting.[5]

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Is a tautology just a nonsense?

Maybe, maybe not. It depends on context:

When you present an analytic statement as an analytic statement, it is not a nonsense.

When you present an analytic statement as a synthetic statement, it is a nonsense.

[2] 重言句, 恆真式
[3] define
[4] or when programming a software
[5] i.e. useful

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Imagine the following scenes.

1.1.3 Scene One

A primary school student wrote 2 + 2 = 5 in his homework. His mathematics teacher told him that 2 + 2 = 5 was incorrect, “Two plus two should equal Four.” In such a context, the statement is, although analytic, not a nonsense.

1.1.4 Scene Two

After 30 years of research, a physicist declared his research result, “Two plus Two equals Four!!!” In such a context, the statement is, although true, a nonsense.

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2008.05.12 Monday \copyright CHK^2

1.1.1 Analytic and Synthetic

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Mathematics is about statements.

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To know what mathematics itself is, we have to realize that there are two kinds of statements: analytic statements and synthetic statements.

For an analytic statement, there is no information about the objective world. Whether an analytic statement is true or not depends on only the meanings of the component words. No real world experience is needed.

For a synthetic statement, there is some information about the objective world. Whether a synthetic statement is true or not depends on not only the meaning of the component words, but also the objective facts of the world.

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For example, consider this statement:

I have passed the exam or I have not.

It is an analytic statement … because you do not have to check my examination result to verify the statement. As long as you know the meanings of “or” and “not”, you know that the statement is always true. But it says nothing about the world.

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Consider another statement:

I have passed the exam.

It is a synthetic statement … because you have to check my examination result to verify the statement. Even if you know the meanings of “or” and “not”, you do not know whether the statement is true or not. But the statement says something about the world.

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2008.05.08 Thursday \copyright CHK^2

3.7.5 Spacetime transform

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3.7 Equivalent Descriptions of an Event
…3.7.1 Vector
…3.7.2 4-vector
…3.7.3 Time evolution – differential equation
…3.7.4 Foresight: spacetime co-ordinate transform
…3.7.5 Envision your future

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3.7.5 Envision your future

Whatever you do now, foresee what your future will be.

For example, when you teach a young boy, imagine how his future is shaped by your now-action.

Whenever you want to shape a future event, find the present-equivalence of that future event, and take action now.

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2008.05.01 Thursday \copyright CHK^2

3.6.7 No persistence needed

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You are working on a big project. You have to work for one month for the project. At every single day of the month, you have to be concentrated. You would have to be persistent to finish the work.

However, if at the beginning of every day, you decide to concentrate for only one day, the task would seem to be lighter. The psychological pressure would be much lower.

Then, at the beginning of another day, you decide to concentrate for only one day again.

Using this mechanism, you just have to be persistent for one day at every time, much easier than to be persistent for the whole month. However, you can still get the benefit of being persistent for the whole month.

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2008.04.19 Saturday \copyright CHK^2

3.6.5 Be always ready

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Ecclesiastes 11:4

If you wait until the wind and the weather are just right, you will never sow anything and never harvest anything.

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Mother Teresa:

The important thing is not to do a lot or to do everything. The important thing is to be ready for anything, at all the times; to be convinced that when serving the poor, we really serve God.

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2008.04.19 Saturday \copyright CHK^2

3.6.4 Now-here-I: the holographic universe

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Sometimes, when I have more than one thing to do, while I am doing one of the things, I keep worrying about the things that I am not doing. This kind of thinking is silly.

When I put my soul in one thing, that thing becomes positive and becomes a positive part of my soul. This gives me enthusiasm to achieve other things effectively.

Just like a holographic film, every part of the film stores all the information needed to reconstruct the whole image. Just like DNA, any single cell in my body contains ALL the DNA data needed to reconstruct my whole body.

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2007.04.19 Saturday \copyright CHK^2

3.6.3 Where are you? Part 2

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3.6.3 You are all of yourselves

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Sometimes, you may wonder whether you can meet your past-self or your future-self through time travelling.

What you do not remember is that you are your own past-self, your own present-self, and your own future-self.

You are all of your-past-present-future-selves.

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2007.04.19 Saturday \copyright CHK^2

3.6.2 Where ARE you?

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3.6.2 Where is your past-self? Where is your future-self?

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They are all now-here. They are all the YOU-now-here.

Now I am a 23-year-old man studying for my master degree. Sometimes, I am re-connected to my 5-year-old mind. I think I am still a 5-year-old boy. I am a 5-year-old who is enjoying university and graduate school now.

Sometimes, I think I am my 90-year-old self, an old man, traveling back in time, to re-live and taste my 23-year-old life, to review my youth.

— Me@2003

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2008.04.16 Wednesday \copyright CHK^2

3.6.1 Foresight: prequel

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Suppose you are an A-Level student. You desire to be a university student.

You should know that

(1997 – 1999; study hard) \Leftrightarrow (1999 – 2002; university student)

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In 1997, I wanted to be a university student. I could not transform myself to be a university student immediately. I could not travel to 1999, becoming a university student at once. However, I knew that, ideally, if I worked hard (and worked smartly) now (1997), I would be able to enter university. In other words, the event “being a university student in 1999” is equivalent to the event “studying hard in 1997”.

I could not control an event in 1999 as it was in my future. But I could control the present-equivalence of that future event.

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2008.04.13 Sunday \copyright CHK^2

3.6 Now-here-I

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David Brower:

Think Globally, Act Locally

小處著手 大處著眼

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An Anglican bishop:

When I was young and free and my imagination had no limits,
I dreamed of changing the world;
As I grew older and wiser I realized the world would not change.

And I decided to shorten my sights somewhat
and change only my country.
But it too seemed immovable.

As I entered my twilight years, in one last desperate attempt,
I sought to change only my family, those closest to me,
but alas they would have none of it.

And now here I lie on my death bed and realize
(perhaps for the first time)
that if only I’d changed myself first,

then by example I may have in
my family and with their encouragement
and support I may have bettered my country,
and who knows I may change the world.

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Mahatma Gandhi:

We must be the change we seek in the world.

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(myself: )

Now-here-I philosophy:

Transform all your goals to a single step that you can work on now and here, by yourself.

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2008.04.12 Saturday \copyright CHK^2

3.5.8 preserve time

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You can’t preserve time by not using it. When time is not used, time would still pass.

You can preserve time only by using it, spending it, investing it, transforming it into something much more valuable.

If you know how to preserve time, you do not fear the passage of time. Instead, you love the passage. As time goes, more good works are getting done.

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2008.04.10 Thursday \copyright CHK^2

當下即是

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If we take eternity to mean not infinite temporal duration but timelessness, then eternal life belongs to those live in present.

— Wittgenstein

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This quotation induces me to have the following thought:

If you concentrate on a meaningful work now and here, you preserve your infinity. (No explanation here.)

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2008.04.09 Wednesday \copyright CHK^2

3.5.7 palm

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To see a world in a grain of sand
And a heaven in a wild
Hold infinity in a palm of your hand
And eternity in an hour

— William Blake

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無人能夠奪去屬於我的一秒鐘.

我永遠存活在那一秒鐘 –

過去, 現在, 未來, 盡收其中.

— Mr. Lee’s 網上思考

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一沙一世界
一花一天堂
掌中握無限
刹那即永恆

— 佛家

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2008.04.07 Monday \copyright CHK^2